Filling 1-cycles, 2-cycle complexity, and torsion growth

Cameron Gates Rudd

Filling 1-cycles, 2-cycle complexity, and torsion growth

Geometric Topology 2026-07-15 v1 Group Theory

Authors: Cameron Gates Rudd

Abstract

For 2-complexes with trivial first Betti number, we study quantitative connections between filling inequalities for 1-cycles, the complexity of the second homology, and the size of the torsion part of the first homology. For 2-complexes whose fundamental groups have property $(\tau)$ with respect to all finite index normal subgroups, we prove a geometric lower bound on the logarithm of the size of the first homology of finite covers.

Keywords

filling inequalities 2-complexes torsion growth property (tau) homology

Cite

@article{arxiv.2607.13736,
  title  = {Filling 1-cycles, 2-cycle complexity, and torsion growth},
  author = {Cameron Gates Rudd},
  journal= {arXiv preprint arXiv:2607.13736},
  year   = {2026}
}